Integrand size = 26, antiderivative size = 41 \[ \int \frac {-1+\log (3 x)}{x \left (1-\log (3 x)+\log ^2(3 x)\right )} \, dx=\frac {\arctan \left (\frac {1-2 \log (3 x)}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (1-\log (3 x)+\log ^2(3 x)\right ) \] Output:
1/3*arctan(1/3*(1-2*ln(3*x))*3^(1/2))*3^(1/2)+1/2*ln(1-ln(3*x)+ln(3*x)^2)
Time = 0.13 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.02 \[ \int \frac {-1+\log (3 x)}{x \left (1-\log (3 x)+\log ^2(3 x)\right )} \, dx=-\frac {\arctan \left (\frac {-1+2 \log (3 x)}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (1-\log (3 x)+\log ^2(3 x)\right ) \] Input:
Integrate[(-1 + Log[3*x])/(x*(1 - Log[3*x] + Log[3*x]^2)),x]
Output:
-(ArcTan[(-1 + 2*Log[3*x])/Sqrt[3]]/Sqrt[3]) + Log[1 - Log[3*x] + Log[3*x] ^2]/2
Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {3039, 25, 1142, 25, 1083, 217, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\log (3 x)-1}{x \left (\log ^2(3 x)-\log (3 x)+1\right )} \, dx\) |
\(\Big \downarrow \) 3039 |
\(\displaystyle \int -\frac {1-\log (3 x)}{\log ^2(3 x)-\log (3 x)+1}d\log (3 x)\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {1-\log (3 x)}{\log ^2(3 x)-\log (3 x)+1}d\log (3 x)\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {1}{2} \int -\frac {1-2 \log (3 x)}{\log ^2(3 x)-\log (3 x)+1}d\log (3 x)-\frac {1}{2} \int \frac {1}{\log ^2(3 x)-\log (3 x)+1}d\log (3 x)\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{2} \int \frac {1}{\log ^2(3 x)-\log (3 x)+1}d\log (3 x)-\frac {1}{2} \int \frac {1-2 \log (3 x)}{\log ^2(3 x)-\log (3 x)+1}d\log (3 x)\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \int \frac {1}{-(2 \log (3 x)-1)^2-3}d(2 \log (3 x)-1)-\frac {1}{2} \int \frac {1-2 \log (3 x)}{\log ^2(3 x)-\log (3 x)+1}d\log (3 x)\) |
\(\Big \downarrow \) 217 |
\(\displaystyle -\frac {1}{2} \int \frac {1-2 \log (3 x)}{\log ^2(3 x)-\log (3 x)+1}d\log (3 x)-\frac {\arctan \left (\frac {2 \log (3 x)-1}{\sqrt {3}}\right )}{\sqrt {3}}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {1}{2} \log \left (\log ^2(3 x)-\log (3 x)+1\right )-\frac {\arctan \left (\frac {2 \log (3 x)-1}{\sqrt {3}}\right )}{\sqrt {3}}\) |
Input:
Int[(-1 + Log[3*x])/(x*(1 - Log[3*x] + Log[3*x]^2)),x]
Output:
-(ArcTan[(-1 + 2*Log[3*x])/Sqrt[3]]/Sqrt[3]) + Log[1 - Log[3*x] + Log[3*x] ^2]/2
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[u_, x_Symbol] :> With[{lst = FunctionOfLog[Cancel[x*u], x]}, Simp[1/lst [[3]] Subst[Int[lst[[1]], x], x, Log[lst[[2]]]], x] /; !FalseQ[lst]] /; NonsumQ[u]
Time = 0.09 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(\frac {\ln \left (1-\ln \left (3 x \right )+\ln \left (3 x \right )^{2}\right )}{2}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (-1+2 \ln \left (3 x \right )\right ) \sqrt {3}}{3}\right )}{3}\) | \(38\) |
default | \(\frac {\ln \left (1-\ln \left (3 x \right )+\ln \left (3 x \right )^{2}\right )}{2}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (-1+2 \ln \left (3 x \right )\right ) \sqrt {3}}{3}\right )}{3}\) | \(38\) |
risch | \(\frac {\ln \left (\ln \left (3 x \right )-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{2}+\frac {i \ln \left (\ln \left (3 x \right )-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{6}+\frac {\ln \left (\ln \left (3 x \right )-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{2}-\frac {i \ln \left (\ln \left (3 x \right )-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{6}\) | \(70\) |
Input:
int((-1+ln(3*x))/x/(1-ln(3*x)+ln(3*x)^2),x,method=_RETURNVERBOSE)
Output:
1/2*ln(1-ln(3*x)+ln(3*x)^2)-1/3*3^(1/2)*arctan(1/3*(-1+2*ln(3*x))*3^(1/2))
Time = 0.07 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.95 \[ \int \frac {-1+\log (3 x)}{x \left (1-\log (3 x)+\log ^2(3 x)\right )} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \log \left (3 \, x\right ) - \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{2} \, \log \left (\log \left (3 \, x\right )^{2} - \log \left (3 \, x\right ) + 1\right ) \] Input:
integrate((-1+log(3*x))/x/(1-log(3*x)+log(3*x)^2),x, algorithm="fricas")
Output:
-1/3*sqrt(3)*arctan(2/3*sqrt(3)*log(3*x) - 1/3*sqrt(3)) + 1/2*log(log(3*x) ^2 - log(3*x) + 1)
Time = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.54 \[ \int \frac {-1+\log (3 x)}{x \left (1-\log (3 x)+\log ^2(3 x)\right )} \, dx=\operatorname {RootSum} {\left (3 z^{2} - 3 z + 1, \left ( i \mapsto i \log {\left (- 3 i + \log {\left (3 x \right )} + 1 \right )} \right )\right )} \] Input:
integrate((-1+ln(3*x))/x/(1-ln(3*x)+ln(3*x)**2),x)
Output:
RootSum(3*_z**2 - 3*_z + 1, Lambda(_i, _i*log(-3*_i + log(3*x) + 1)))
Time = 0.11 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.90 \[ \int \frac {-1+\log (3 x)}{x \left (1-\log (3 x)+\log ^2(3 x)\right )} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \log \left (3 \, x\right ) - 1\right )}\right ) + \frac {1}{2} \, \log \left (\log \left (3 \, x\right )^{2} - \log \left (3 \, x\right ) + 1\right ) \] Input:
integrate((-1+log(3*x))/x/(1-log(3*x)+log(3*x)^2),x, algorithm="maxima")
Output:
-1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*log(3*x) - 1)) + 1/2*log(log(3*x)^2 - l og(3*x) + 1)
\[ \int \frac {-1+\log (3 x)}{x \left (1-\log (3 x)+\log ^2(3 x)\right )} \, dx=\int { \frac {\log \left (3 \, x\right ) - 1}{{\left (\log \left (3 \, x\right )^{2} - \log \left (3 \, x\right ) + 1\right )} x} \,d x } \] Input:
integrate((-1+log(3*x))/x/(1-log(3*x)+log(3*x)^2),x, algorithm="giac")
Output:
integrate((log(3*x) - 1)/((log(3*x)^2 - log(3*x) + 1)*x), x)
Time = 27.47 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.90 \[ \int \frac {-1+\log (3 x)}{x \left (1-\log (3 x)+\log ^2(3 x)\right )} \, dx=\frac {\ln \left ({\ln \left (3\,x\right )}^2-\ln \left (3\,x\right )+1\right )}{2}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,\left (2\,\ln \left (3\,x\right )-1\right )}{3}\right )}{3} \] Input:
int((log(3*x) - 1)/(x*(log(3*x)^2 - log(3*x) + 1)),x)
Output:
log(log(3*x)^2 - log(3*x) + 1)/2 - (3^(1/2)*atan((3^(1/2)*(2*log(3*x) - 1) )/3))/3
Time = 0.16 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.88 \[ \int \frac {-1+\log (3 x)}{x \left (1-\log (3 x)+\log ^2(3 x)\right )} \, dx=-\frac {\sqrt {3}\, \mathit {atan} \left (\frac {2 \,\mathrm {log}\left (3 x \right )-1}{\sqrt {3}}\right )}{3}+\frac {\mathrm {log}\left (\mathrm {log}\left (3 x \right )^{2}-\mathrm {log}\left (3 x \right )+1\right )}{2} \] Input:
int((-1+log(3*x))/x/(1-log(3*x)+log(3*x)^2),x)
Output:
( - 2*sqrt(3)*atan((2*log(3*x) - 1)/sqrt(3)) + 3*log(log(3*x)**2 - log(3*x ) + 1))/6